Progress on Parallel Chip-Firing
Ziv Scully
MIT PRIMES
May 21, 2011
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
1 / 18
Motivation
Simple rules
“Obvious” patterns which are difficult to prove, or even wrong
Potential connections to other fields of mathematics and science
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
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Graphs
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
3 / 18
Graphs
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
3 / 18
Graphs
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
3 / 18
Graphs
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
3 / 18
Graphs
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
3 / 18
The Parallel Chip-Firing Game
Played on a graph
Assign a number of chips to each vertex
On each turn:
If a vertex has at least as many chips as neighbors, it fires
Otherwise, we say it waits
When a vertex fires, it gives one chip to each of its neighbors
Happens for all vertices in parallel
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
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The Parallel Chip-Firing Game
c 0
b 1
d 0
abc def gh
51000200
a 5
g 0
f 2
e 0
h 0
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 0
b 1
d 0
abc def gh
51000200
a 5
g 0
f 2
e 0
h 0
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 0
b 2
d 1
abc def gh
51000200
22012010
a 2
g 1
f 0
e 2
h 0
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 0
b 2
d 1
abc def gh
51000200
22012010
a 2
g 1
f 0
e 2
h 0
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 1
b 0
d 1
abc def gh
51000200
22012010
30112100
a 3
g 0
f 1
e 2
h 0
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 1
b 0
d 1
abc def gh
51000200
22012010
30112100
a 3
g 0
f 1
e 2
h 0
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 1
b 1
d 2
abc def gh
51000200
22012010
30112100
01123100
a 0
g 0
f 1
e 3
h 0
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 1
b 1
d 2
abc def gh
51000200
22012010
30112100
01123100
a 0
g 0
f 1
e 3
h 0
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 2
b 1
d 0
abc def gh
51000200
22012010
30112100
01123100
21200201
a 2
g 0
f 2
e 0
h 1
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 2
b 1
d 0
abc def gh
51000200
22012010
30112100
01123100
21200201
a 2
g 0
f 2
e 0
h 1
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 0
b 2
d 1
abc def gh
51000200
22012010
30112100
01123100
21200201
22012010
Ziv Scully (MIT PRIMES)
a 2
g 1
f 0
e 2
h 0
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
The Parallel Chip-Firing Game
c 0
b 2
d 1
abc def gh
51000200
22012010
30112100
01123100
21200201
22012010
Ziv Scully (MIT PRIMES)
a 2
g 1
f 0
e 2
h 0
Progress on Parallel Chip-Firing
May 21, 2011
5 / 18
Basic Properties
All games are eventually periodic
All vertices fire the same number of times in a period
In a periodic-1 position, either all vertices fire or all vertices wait
Period > 2 needs a cycle
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
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Notation
σ(t) is the position after taking t turns, starting with position σ(0)
σv (t) is the number of chips on vertex v in position σ(t)
Φv (t) is the number of v ’s neighbors that fire at time t; v gets one
chip from each
Fv (t) is 1 if v fires at time t and 0 otherwise
c is the total number of chips in a position
If G is a graph, V (G ) is its vertex set and E (G ) is its edge set
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
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Outline of Literature
Bitar’s conjecture: maximum period ≤ number of vertices
Bitar and Goles: Trees have period 1 or 2
Kiwi et al.: Bitar’s conjecture is false!
Dall’Asta: Period on Cn divides n
Levine: Period on Kn ≤ n
Jiang: Period on Ka,b ≤ 2 min(a, b)
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
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Periodic or Not?
c 0
c 0
b 1
d 0
a 5
g 0
b 2
d 1
a 2
f 2
e 0
h 0
f 0
e 2
Period 4
(not periodic)
Ziv Scully (MIT PRIMES)
g 1
Periodic-4
h 0
Progress on Parallel Chip-Firing
May 21, 2011
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Periodic-2 Positions
Theorem (Characterization of periodic-2 positions)
A position σ(t) on graph G is periodic-2 if and only if for all v ∈ V (G ),
deg(v ) ≤ σv (t) + Φv (t) ≤ 2 deg(v ) − 1.
Proof.
When the period is 2, vertices alternate between firing and waiting. The
above inequality is true if and only if v is about to switch states.
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
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Understanding Trees
0
0
0
0
7
0
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8
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Progress on Parallel Chip-Firing
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May 21, 2011
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Understanding Trees
1
1
0
1
3
1
0
Ziv Scully (MIT PRIMES)
1
4
1
0
0
0
1
0
Progress on Parallel Chip-Firing
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May 21, 2011
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Understanding Trees
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1
5
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Ziv Scully (MIT PRIMES)
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2
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1
0
Progress on Parallel Chip-Firing
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Understanding Trees
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1
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Ziv Scully (MIT PRIMES)
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5
2
0
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1
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Progress on Parallel Chip-Firing
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Understanding Trees
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1
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4
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Ziv Scully (MIT PRIMES)
2
1
2
0
0
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1
0
Progress on Parallel Chip-Firing
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Understanding Trees
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Ziv Scully (MIT PRIMES)
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1
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Progress on Parallel Chip-Firing
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May 21, 2011
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Understanding Trees
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Ziv Scully (MIT PRIMES)
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Progress on Parallel Chip-Firing
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May 21, 2011
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Understanding Trees
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Progress on Parallel Chip-Firing
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1
May 21, 2011
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Understanding Trees
Period 1
0
Period 2
1
0
1
3
0
0
Ziv Scully (MIT PRIMES)
2
0
3
0
0
0
1
0
Progress on Parallel Chip-Firing
4
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0
May 21, 2011
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Understanding Trees
Period 1
0
Period 2
0
0
1
3
0
0
Ziv Scully (MIT PRIMES)
0
4
3
0
0
1
0
1
Progress on Parallel Chip-Firing
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May 21, 2011
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Understanding Trees
Period 1
0
Period 2
1
0
1
3
0
0
Ziv Scully (MIT PRIMES)
2
0
3
0
0
0
1
0
Progress on Parallel Chip-Firing
4
0
0
May 21, 2011
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Understanding Trees
Theorem (Number of chips on a tree determines period)
If a game on a tree graph G has c chips, its eventual period is 2 if and
only if |E (G )| ≤ c ≤ 2|E (G )| − 1.
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
12 / 18
Understanding Trees
Theorem (Number of chips on a tree determines period)
If a game on a tree graph G has c chips, its eventual period is 2 if and
only if |E (G )| ≤ c ≤ 2|E (G )| − 1.
Proof.
If the period is n, then for some time t, σ(t) will be periodic-n.
If n = 1:
σv (t) ≤ deg(v ) − 1
deg(v ) ≤ σv (t)
c ≤ |E (G )| − 1
If n = 2:
2|E (G )| ≤ c
deg(v ) ≤ σv (t) + Φv (t) ≤ 2 deg(v ) − 1
X Φv (t) + Φv (t + 1)
2|E (G )| ≤ c +
≤ 3|E (G )| − 1
2
|E (G )| ≤ c ≤ 2|E (G )| − 1
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
12 / 18
Firing Patterns
String of 1s and 0s indicating firing and waiting, respectively
Classification
Alternating: (1, 0)
Sparse: not alternating, two types
Sparsely firing: never fires twice in a row
Sparsely waiting: never waits twice in a row
Clumpy: neither sparse nor alternating
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
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Motors
A special vertex with a fixed firing pattern
Doesn’t care about receiving chips
Natural motors
Subgraphs that follow normal chip firing rules
One key vertex behaves like a motor
Receiving external chips doesn’t change its firing pattern
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
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Motorized Trees
some
graph
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
15 / 18
Motorized Trees
some
graph
Ziv Scully (MIT PRIMES)
motor
Progress on Parallel Chip-Firing
May 21, 2011
15 / 18
Motorized Trees
some
graph
motor
Theorem (Periodic behavior of trees with one sparse motor)
If motor m in tree graph G is sparse, then for all v ∈ V (G ) at any periodic
time t, Fv (t) = Fm (t − d), where d is the distance from m to v .
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
15 / 18
Motorized Trees
some
graph
motor
Theorem (Periodic behavior of trees with one sparse motor)
If motor m in tree graph G is sparse, then for all v ∈ V (G ) at any periodic
time t, Fv (t) = Fm (t − d), where d is the distance from m to v .
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
15 / 18
Motorized Trees
some
graph
motor
Theorem (Periodic behavior of trees with one sparse motor)
If motor m in tree graph G is sparse, then for all v ∈ V (G ) at any periodic
time t, Fv (t) = Fm (t − d), where d is the distance from m to v .
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
15 / 18
Motorized Trees
some
graph
motor
Theorem (Periodic behavior of trees with one sparse motor)
If motor m in tree graph G is sparse, then for all v ∈ V (G ) at any periodic
time t, Fv (t) = Fm (t − d), where d is the distance from m to v .
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
15 / 18
Motorized Trees
some
graph
motor
Theorem (Periodic behavior of trees with one sparse motor)
If motor m in tree graph G is sparse, then for all v ∈ V (G ) at any periodic
time t, Fv (t) = Fm (t − d), where d is the distance from m to v .
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
15 / 18
Motorized Trees
some
graph
motor
Theorem (Periodic behavior of trees with one sparse motor)
If motor m in tree graph G is sparse, then for all v ∈ V (G ) at any periodic
time t, Fv (t) = Fm (t − d), where d is the distance from m to v .
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
15 / 18
Motorized Trees
some
graph
motor
Theorem (Periodic behavior of trees with one sparse motor)
If motor m in tree graph G is sparse, then for all v ∈ V (G ) at any periodic
time t, Fv (t) = Fm (t − d), where d is the distance from m to v .
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
15 / 18
Constructing Natural Sparse and Alternating Motors
2 (each)
0
1
1
1
1
Ziv Scully (MIT PRIMES)
5
Progress on Parallel Chip-Firing
1
May 21, 2011
16 / 18
Constructing Natural Sparse and Alternating Motors
2
0
1
1
1
1
5
1
(0,
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
16 / 18
Constructing Natural Sparse and Alternating Motors
0
1
2
1
1
1
5
1
(0, 0,
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
16 / 18
Constructing Natural Sparse and Alternating Motors
1
1
0
1
2
1
5
1
(0, 0, 0,
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
16 / 18
Constructing Natural Sparse and Alternating Motors
1
1
1
1
0
1
5
2
(0, 0, 0, 0,
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
16 / 18
Constructing Natural Sparse and Alternating Motors
1
1
1
1
1
1
10
0
(0, 0, 0, 0, 1,
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
16 / 18
Constructing Natural Sparse and Alternating Motors
1
1
1
1
1
2
0
1
(0, 0, 0, 0, 1, 0,
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
16 / 18
Constructing Natural Sparse and Alternating Motors
1
1
1
2
1
0
5
1
(0, 0, 0, 0, 1, 0, 0,
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
16 / 18
Constructing Natural Sparse and Alternating Motors
1
2
1
0
1
1
5
1
(0, 0, 0, 0, 1, 0, 0, 0
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
16 / 18
Constructing Natural Sparse and Alternating Motors
2
0
1
1
1
1
5
1
(0, 0, 0, 0, 1, 0, 0, 0)
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
16 / 18
Further Questions
Can a vertex have a clumpy firing pattern in a period?
Can every vertex firing be traced back to a “driving cycle”?
If a graph has a possible period of length mp for some prime p, must
the graph have a cycle of length np?
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
17 / 18
Acknowledgments
Mentor, Yan Zhang, MIT
Collaborator, Damien Jiang, MIT
MIT Program for Research in Mathematics, Engineering and Science
Dr. Anne Fey, TU Delft
Dr. Tanya Khovanova, MIT
Dr. Lionel Levine, MIT
Ziv Scully (MIT PRIMES)
Progress on Parallel Chip-Firing
May 21, 2011
18 / 18